Page 437 - Contributed Paper Session (CPS) - Volume 4

CPS2526 Holger Cevallos-Valdiviezo et al.

Least trimmed squares estimators for
functional principal component analysis
3
1
2
Holger Cevallos-Valdiviezo , Stefan Van Aelst , Matias Salibian-Barrera
1 Escuela Superior Politécnica del Litoral (ESPOL), Facultad de Ciencias Naturales y Matemáticas
(FCNM), Km 30.5 Vía Perimetral, Guayaquil, Ecuador
2 KU Leuven, Department of Mathematics, Section of Statistics, Celestijnenlaan 200B B-3001,
Leuven, Belgium
3 The University of British Columbia, Department of Statistics, 3182 Earth Sciences Building
(ESB), Vancouver, BC V6T 1Z4, Canada

Abstract
Classical functional principal component analysis can yield erroneous
approximations in presence of outliers. To reduce the infuence of atypical data
we propose two methods based on trimming: a multivariate least trimmed
squares (LTS) estimator and its coordinatewise variant. The multivariate LTS
minimizes the multivariate scale corresponding to hsubsets of curves while the
coordinatewise version uses univariate LTS scale estimators. Consider a
general setup in which observations are realizations of a random element on
a separable Hilbert space H. For a fixed dimension q, we aim to robustly
estimate the q dimensional linear space in H that gives the best approximation
to the functional data. Our estimators use smoothing to first represent
irregularly spaced curves in a high-dimensional space and then calculate the
LTS solution on these multivariate data. The solution of the multivariate data
is subsequently mapped back onto H. Poorly fitted observations can therefore
be flagged as outliers. Simulations and real data applications show that our
estimators yield competitive results when compared to existing methods when
a minority of observations is contaminated. When a majority of the curves is
contaminated at some positions along its trajectory coordinatewise methods
like Coordinatewise LTS are preferred over multivariate LTS and other
multivariate methods since they break down in this case.

Keywords
Functional Data Analysis; Robust Methods

1. Introduction
For a fixed dimension q, functional principal component analysis (FPCA) aims
to estimate the q dimensional linear space that gives the best approximation
to the functional data. For instance, the classical approach for FPCA has the
property of providing optimal approximations in the L2 sense. However, this
approach is very sensitive to abnormal functional data. To reduce the influence
of outliers we propose two methods based on trimming: a multivariate least

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